In this paper we are concerned with resolvent estimates for the Laplacian ∆ in Euclidean spaces. Uniform resolvent estimates for ∆ were shown by Kenig, Ruiz and Sogge [32] who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp L p -L q bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp L p -L q resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner-Riesz operators of negative index.2010 Mathematics Subject Classification. 35B45, 42B15.

Abstract. We study uniform Sobolev inequalities for the second order differential operators P pDq of non-elliptic type. For d ě 3 we prove that the Sobolev type estimate }u} L q pR d q ď C}P pDqu} L p pR d q holds with C independent of the first order and the constant terms of P pDq if and only if 1{p´1{q " 2{d and. We also obtain restricted weak type endpoint estimates for the critical pp, qq " pd´2 q. As a consequence, the result extends the class of functions for which the unique continuation for the inequality |P pDqu| ď |V u| holds.

In this paper, we consider the Lamé operator −∆ * and study resolvent estimate, uniform Sobolev estimate, and Carlman estimate for −∆ * . First, we obtain sharp L p -L q resolvent estimates for −∆ * for admissible p, q. This extends the particular case q = p p−1 due to Cossetti [6]. Secondly, we show failure of uniform Sobolev estimate and Carlman estimate for −∆ * . Our approach is different from that in [6] which relies heavily on the Helmholtz decomposition. Instead, we directly analyze the Fourier multiplier of the resolvent. This direct approach allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp L p -L q bounds for the resolvent of −∆ * . Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator −∆ * even though the uniform resolvent estimates for −∆ * are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian ∆ due to Kenig, Ruiz, and Sogge [21] in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carlman estimates for ∆. We also describe locations of the L q -eigenvalues of −∆ * + V with complex potential V by making use of the sharp L p -L q resolvent estimates for −∆ * .

Let P(D) be the Laplacian ∆, or the wave operator . The following type of Carleman estimate is known to be true on a certain range of p, q:The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge [15] and Jeong-Kwon-Lee [13]. The range of p, q for which the uniform Sobolev type estimates hold was completely characterized for the second order differential operators with nondegenerate principal part. But the optimal range of p, q for which the Carleman estimate holds has not been clarified before. When P(D) = ∆, , or the heat operator, we obtain a complete characterization of the admissible p, q for the aforementioned type of Carleman estimate. For this purpose we investigate L p -L q boundedness of related multiplier operators. As applications, we also obtain some unique continuation results.